Quantum-state engineering with Josephson-junction devicesfy.chalmers.se/~delsing/QI/Makhlin-RMP-01.pdf · 2003. 1. 15. · Josephson junction or the ﬂux in a ring geometry near

REVIEWS OF MODERN PHYSICS, VOLUME 73, APRIL 2001

Quantum-state engineering with Josephson-junction devices

Yuriy Makhlin

Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe,Germanyand Landau Institute for Theoretical Physics, Kosygin st. 2, 117940 Moscow, Russia

Gerd Schön

Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe,Germanyand Forschungszentrum Karlsruhe, Institut für Nanotechnologie, D-76021 Karlsruhe,Germany

Alexander Shnirman

Institut für Theoretische Festkörperphysik, Universität Karlsruhe,D-76128 Karlsruhe, Germany

(Published 8 May 2001)

Quantum-state engineering, i.e., active control over the coherent dynamics of suitablequantum-mechanical systems, has become a fascinating prospect of modern physics. With conceptsdeveloped in atomic and molecular physics and in the context of NMR, the field has been stimulatedfurther by the perspectives of quantum computation and communication. Low-capacitance Josephsontunneling junctions offer a promising way to realize quantum bits (qubits) for quantum informationprocessing. The article reviews the properties of these devices and the practical and fundamentalobstacles to their use. Two kinds of device have been proposed, based on either charge or phase (flux)degrees of freedom. Single- and two-qubit quantum manipulations can be controlled by gate voltagesin one case and by magnetic fields in the other case. Both kinds of device can be fabricated withpresent technology. In flux qubit devices, an important milestone, the observation of superpositions ofdifferent flux states in the system eigenstates, has been achieved. The Josephson charge qubit has evendemonstrated coherent superpositions of states readable in the time domain. There are two majorproblems that must be solved before these devices can be used for quantum information processing.One must have a long phase coherence time, which requires that external sources of dephasing beminimized. The review discusses relevant parameters and provides estimates of the decoherence time.Another problem is in the readout of the final state of the system. This issue is illustrated with apossible realization by a single-electron transistor capacitively coupled to the Josephson device, butgeneral properties of measuring devices are also discussed. Finally, the review describes how the basicphysical manipulations on an ideal device can be combined to perform useful operations.

CONTENTS

I. Introduction 357II. Josephson Charge Qubit 359

A. Superconducting charge box as a quantum bit 359B. Charge qubit with tunable coupling 361C. Controlled interqubit coupling 362D. Experiments with Josephson charge qubits 364E. Adiabatic charge manipulations 365

III. Qubits Based on the Flux Degree of Freedom 366A. Josephson flux (persistent current) qubits 367B. Coupling of flux qubits 369C. ‘‘Quiet’’ superconducting phase qubits 369D. Switches 371

IV. Environment and Dissipation 371A. Identifying the problem 371B. Spin-boson model 372C. Several fluctuating fields and many qubits 374D. Dephasing in charge qubits 374E. Dephasing in flux qubits 376

V. The Quantum Measurement Process 377A. General concept of quantum measurements 377B. Single-electron transistor as a quantum

electrometer 378

0034-6861/2001/73(2)/357(44)/$28.80 357

C. Density matrix and description of measurement 380D. Master equation 381E. Hamiltonian-dominated regime 382F. Detector-dominated regime 385G. Flux measurements 386H. Efficiency of the measuring device 386I. Statistics of the current and the noise spectrum 388J. Conditional master equation 389

VI. Conclusions 390Acknowledgments 392Appendix A: An Ideal Model 392

1. The model Hamiltonian 3922. Preparation of the initial state 3933. Single-qubit operations 3934. Two-qubit operations 393

Appendix B: Quantum Logic Gates and Quantum Algorithms 3931. Single- and two-qubit gates 3932. Quantum Fourier transformation 3943. Quantum computation and optimization 394

Appendix C: Charging Energy of a Qubit Coupled to a Set 395Appendix D: Derivation of the Master Equation 395References 397

I. INTRODUCTION

The interest in ‘‘macroscopic’’ quantum effects in low-capacitance Josephson-junction circuits has persisted for

©2001 The American Physical Society

358 Makhlin, Schön, and Shnirman: Quantum-state engineering

many years. One of the motivations was to test whetherthe laws of quantum mechanics applied in macroscopicsystems, in a Hilbert space spanned by macroscopicallydistinct states (Leggett, 1987). The degrees of freedomstudied were the phase difference of the superconduct-ing order parameter across a junction or the flux in asuperconducting quantum interference device (SQUID)ring geometry. Various quantum phenomena, such asmacroscopic quantum tunneling and resonance tunnel-ing, were demonstrated (see, for example, Voss andWebb, 1981; Martinis et al., 1987; Rouse et al., 1995). Onthe other hand, despite experimental efforts (e.g.,Tesche, 1990), coherent oscillations of the flux betweentwo macroscopically distinct states (macroscopic quan-tum coherence) had not been observed.

The field received new attention recently, after it wasrecognized that suitable Josephson devices might serveas quantum bits (qubits) in quantum information devicesand that quantum logic operations1 could be performedby controlling gate voltages or magnetic fields (see, forexample, Bouchiat, 1997; Shnirman et al., 1997; Averin,1998; Ioffe et al., 1999; Makhlin et al., 1999; Mooij et al.,1999; Nakamura et al., 1999). In this context, as well asfor other conceivable applications of quantum-state en-gineering, the experimental milestones are the observa-tion of quantum superpositions of macroscopically dis-tinct states, of coherent oscillations, and of entangledquantum states of several qubits. For Josephson devicesthe first successful experiments have been performed.These systems can be fabricated by established litho-graphic methods, and the control and measurementtechniques are quite advanced. They further exploit thecoherence of the superconducting state, which helps toachieve sufficiently long phase coherence times.

Two alternative realizations of quantum bits havebeen proposed, based on either charge or phase (flux)degrees of freedom. In the former, the charge in low-capacitance Josephson junctions is used as a quantumdegree of freedom, with basis states differing by thenumber of Cooper-pair charges on an island. These de-vices combine the coherence of Cooper-pair tunnelingwith the control mechanisms developed for single-charge systems and Coulomb-blockade phenomena. Themanipulations can be accomplished by switching gatevoltages (Shnirman et al., 1997); designs with controlledinterqubit couplings were proposed (Averin, 1998;Makhlin et al., 1999). Experimentally, the coherent tun-neling of Cooper pairs and the related properties ofquantum-mechanical superpositions of charge stateshave been demonstrated (Bouchaiat, 1997; Nakamuraet al., 1997). Most spectacular are recent experiments ofNakamura et al. (1999) in which the quantum-coherentoscillations of a Josephson charge qubit prepared in a

1Since computational applications are widely discussed, wefrequently employ here and below the terminology of quantuminformation theory, referring to a two-state quantum system asa qubit and denoting unitary manipulations of its quantumstate as quantum logic operations or gates.

Rev. Mod. Phys., Vol. 73, No. 2, April 2001

superposition of eigenstates were observed in the timedomain. We describe these systems, concepts, and re-sults in Sec. II.

The alternative realization is based on the phase of aJosephson junction or the flux in a ring geometry near adegeneracy point as a quantum degree of freedom (see,for example, Ioffe et al., 1999; Mooij et al., 1999). In ad-dition to the earlier experiments, in which macroscopicquantum tunneling had been observed (Voss and Webb,1981; Martinis et al., 1987; Rouse et al., 1995), the groupsin Delft and Stony Brook (Friedman et al., 2000; van derWal et al., 2000) recently demonstrated by spectroscopicmeasurements the flux qubit’s eigenenergies; they ob-served eigenstates that are superpositions of differentflux states, and new efforts are being made to observethe coherent oscillation of the flux between degeneratestates (Cosmelli et al., 1998; Mooij et al., 1999; Friedmanet al., 2000). We shall discuss the quantum properties offlux qubits in Sec. III.

To make use of the quantum coherent time evolutionit is crucial to find systems with intrinsically long phasecoherence times and to minimize external sources ofdephasing. The latter can never be avoided completelysince, in order to perform the necessary manipulations,one has to couple to the qubits, for instance, by attach-ing external leads. Along the same channels as the signal(e.g., gate voltages) noise also enters the system. How-ever, by operating at low temperatures and choosingsuitable coupling parameters, one can keep thesedephasing effects at an acceptable level. We provide es-timates of the phase coherence time in Sec. IV.

In addition to controlled manipulations of qubits,quantum measurement processes are needed, for ex-ample, to read out the final state of the system. In ourquantum mechanics courses we learned to express themeasurement process as a ‘‘wave-function collapse,’’ i.e.,as a nonunitary projection, which reduces the quantumstate of the qubit to one of the possible eigenstates ofthe observed quantity with state-dependent probabili-ties. However, in reality any measurement is performedby a device that itself is realized by a physical system,suitably coupled to the measured quantum system andwith a macroscopic readout variable. Its presence, ingeneral, disturbs the quantum manipulations. Thereforethe dissipative processes that accompany the measure-ment should be switched on only when needed.

An example is provided by a normal-state single-electron transistor (SET) coupled capacitively to asingle-Cooper-pair box. This system is widely used as anelectrometer in classical single-charge systems. We de-scribe in Sec. V how a SET can also be used to read outthe quantum state of a charge qubit. For this purpose westudy the time evolution of the coupled system’s densitymatrix (Shnirman and Schön, 1998). During quantummanipulations of the qubit the transport voltage of theSET is turned off, in which case it acts only as an extracapacitor. To perform the measurement the transportvoltage is turned on. In this stage the dissipative currentthrough the transistor rapidly dephases the state of thequbit. This current also provides the macroscopic read-

359Makhlin, Schön, and Shnirman: Quantum-state engineering

out signal for the quantum state of the qubit. However,it requires a longer ‘‘measurement time’’ until the noisysignal resolves different qubit states. Finally, on the stilllonger ‘‘mixing time’’ scale, the measurement processitself destroys the information about the initial quantumstate.

Many results and observations made in the context ofthe normal-state single-electron transistor also apply toother physical systems, e.g., a superconducting SET(SSET) coupled to a charge qubit (Averin, 2000b; Cottetet al., 2000) or a dc SQUID monitoring as a quantummagnetometer the state of a flux qubit (see, for example,Mooij et al., 1999; Averin, 2000b; Friedman et al., 2000).The results can also be compared to the nonequilibriumdephasing processes discussed theoretically (Aleineret al., 1997; Gurvitz, 1997; Levinson, 1997) and demon-strated experimentally by Buks et al. (1998).

One of the motivations for quantum-state engineeringwith Josephson devices is their potential application aslogic devices and quantum computing. By exploiting themassive parallelism of the coherent evolution of super-positions of states, quantum computers could performcertain tasks that no classical computer could do in ac-ceptable times (Bennett, 1995; DiVincenzo, 1995;Barenco, 1996; Aharonov, 1998). In contrast to the de-velopment of physical realizations of qubits and gates,i.e., the ‘‘hardware,’’ the theoretical concepts of quan-tum computing, the ‘‘software,’’ are already rather ad-vanced. As an introduction, and in order to clearly de-fine the goals, we present in Appendix A an ideal modelHamiltonian with sufficient control to perform all theneeded manipulations. (We note that the Josephson-junction devices come rather close to this ideal model.)In Appendix B we show by a few representative ex-amples how these manipulations can be combined foruseful computations.

Various other physical systems have been suggestedas possible realizations of qubits and gates. They arediscussed in much detail in a recent Fortschritte derPhysik special issue entitled Experimental Proposals forQuantum Computation (Braunstein and Lo, 2000). Insome systems quantum manipulations of a few qubitshave already been demonstrated experimentally. Theseinclude ions in electromagnetic traps manipulated by la-ser irradiation (Cirac and Zoller, 1995; Monroe et al.,1995), nuclear magnetic resonance (NMR) on ensemblesof molecules in liquids (Cory et al., 1997; Gershenfeldand Chuang, 1997) and cavity QED systems (Turchetteet al., 1995). In comparison, solid-state devices, includingthe mentioned Josephson systems, are more easily em-bedded in electronic circuits and scaled up to large reg-isters. Ultrasmall quantum dots with discrete levels and,in particular, spin degrees of freedom embedded innanostructured materials are candidates as well. Theycan be manipulated by tuning potentials and barriers(Kane, 1998; Loss and DiVincenzo, 1998). Because ofthe difficulties of controlled fabrication, their experi-mental realization is still at a very early stage.


II. JOSEPHSON CHARGE QUBIT

A. Superconducting charge box as a quantum bit

In this section we describe the properties of low-capacitance Josephson junctions, in which the chargingenergy dominates over the Josephson coupling energy,and discuss how they can be manipulated in a quantum-coherent fashion. Under suitable conditions they pro-vide physical realizations of qubits with two states dif-fering by one Cooper pair charge on a small island. Thenecessary one-bit and two-bit gates can be performed bycontrolling applied gate voltages and magnetic fields.Different designs will be presented that differ not onlyin complexity, but also in the accuracy and flexibility ofthe manipulations.

The simplest Josephson-junction qubit is shown inFig. 1. It consists of a small superconducting island(‘‘box’’) with n excess Cooper-pair charges (relative tosome neutral reference state), connected by a tunneljunction with capacitance CJ and Josephson coupling en-ergy EJ to a superconducting electrode. A control gatevoltage Vg is coupled to the system via a gate capacitorCg . Suitable values of the junction capacitance, whichcan be fabricated routinely by present-day technologies,are in the range of femtofarad and below, CJ1 K. The Josephson coupling en-ergy EJ is proportional to the critical current of the Jo-sephson junction (see, for example, Tinkham, 1996).Typical values considered here are in the range of 100mK.

We choose a material such that the superconductingenergy gap D is the largest energy in the problem, largereven than the single-electron charging energy. In this

2Throughout this review we frequently use temperature unitsfor energies.

FIG. 1. A Josephson charge qubit in its simplest design formedby a superconducting single-charge box.


case quasiparticle tunneling is suppressed at low tem-peratures, and a situation can be reached in which noquasiparticle excitation is found on the island.3 Underthese conditions only Cooper pairs tunnel—coherently—in the superconducting junction, and thesystem is described by the Hamiltonian

H54EC~n2ng!22EJ cos Q . (2.1)Here n is the number operator of (excess) Cooper-paircharges on the island, and Q, the phase of the supercon-ducting order parameter of the island, is its quantum-mechanical conjugate, n52i\ ]/](\Q). The dimen-sionless gate charge, ng[CgVg/2e , accounts for theeffect of the gate voltage and acts as a control param-eter. Here we consider systems in which the chargingenergy is much larger than the Josephson coupling en-ergy, EC@EJ . In this regime a convenient basis isformed by the charge states, parametrized by the num-ber of Cooper pairs n on the island. In this basis theHamiltonian (2.1) reads

H5(n

H 4EC~n2ng!2un&^nu2

12

EJ~ un&^n11u1un11&^nu!J . (2.2)For most values of ng the energy levels are dominated

by the charging part of the Hamiltonian. However, whenng is approximately half-integer and the charging ener-gies of two adjacent states are close to each other (e.g.,at Vg5Vdeg[e/Cg), the Josephson tunneling mixes themstrongly (see Fig. 2). We concentrate on such a voltagerange near a degeneracy point where only two chargestates, say n50 and n51, play a role, while all othercharge states, having a much higher energy, can be ig-nored. In this case the superconducting charge box (2.1)reduces to a two-state quantum system (qubit) with aHamiltonian that can be written in spin-12 notation as

Hctrl5212

Bzŝz212

Bxŝx . (2.3)

The charge states n50 and n51 correspond to the spinbasis states u↑&[(01) and u↓&[(10), respectively. Thecharging energy splitting, which is controlled by the gatevoltage, corresponds in spin notation to the z compo-nent of the magnetic field,

Bz[dEch[4EC~122ng!, (2.4)

3In the ground state the superconducting state is totallypaired, which requires an even number of electrons on theisland. A state with an odd number of electrons necessarilycosts an extra quasiparticle energy D and is exponentially sup-pressed at low T . This ‘‘parity effect’’ has been established inexperiments below a crossover temperature T*'D/(kB ln Neff), where Neff is the number of electrons in thesystem near the Fermi energy (Tuominen et al., 1992; Lafargeet al., 1993; Schön and Zaikin, 1994; Tinkham, 1996). For asmall island, T* is typically one order of magnitude lower thanthe superconducting transition temperature.


while the Josephson energy provides the x componentof the effective magnetic field,

Bx[EJ . (2.5)

For later convenience we rewrite the Hamiltonian as

Hctrl5212

DE~h!~cos h sz1sin h sx!, (2.6)

where the mixing angle h[tan21(Bx /Bz) determines thedirection of the effective magnetic field in the x-z plane,and the energy splitting between the eigenstates isDE(h)5ABx21Bz25EJ /sin h. At the degeneracy point,h5p/2, it reduces to EJ . The eigenstates are denoted inthe following as u0& and u1&. They depend on the gatecharge ng as

u0&5cosh

2u↑&1sin

h

2u↓&,

u1&52sinh

2u↑&1cos

h

2u↓&. (2.7)

We can further express the Hamiltonian in the basisof eigenstates. To avoid confusion we introduce a sec-ond set of Pauli matrices r that operate in the basis u0&and u1&, while reserving the operators s for the basis ofcharge states u↑& and u↓&. By definition the Hamiltonianthen becomes

H5212

DE~h!rz . (2.8)

The Hamiltonian (2.3) is similar to the ideal single-qubit model (A1) presented in Appendix A. Ideally thebias energy (the effective magnetic field in the z direc-tion) and the tunneling amplitude (the field in the x di-rection) are controllable. However, at this stage we cancontrol—by the gate voltage—only the bias energy,while the tunneling amplitude has a constant value setby the Josephson energy. Nevertheless, by switching thegate voltage we can perform the required one-bit opera-tions (Shnirman et al., 1997). If, for example, onechooses the idle state far to the left from the degeneracypoint, the eigenstates u0& and u1& are close to u↑& and

FIG. 2. The charging energy of a superconducting electron boxis shown as a function of the gate charge ng for different num-bers of extra Cooper pairs n on the island (dashed parabolas).Near degeneracy points the weaker Josephson coupling mixesthe charge states and modifies the energy of the eigenstates(solid lines). In the vicinity of these points the system effec-tively reduces to a two-state quantum system.


u↓&, respectively. Then switching the system suddenly tothe degeneracy point for a time Dt and back produces arotation in spin space,

U1-bit~a!5expS i a2 sxD5S cos a2 i sin a2i sin

a

2cos

a

2

D , (2.9)where a5EJDt/\ . Depending on the value of Dt , a spinflip can be produced or, starting from u0&, a superposi-tion of states with any chosen weights can be reached.[This is exactly the operation performed in the experi-ments of Nakamura et al., (1999); see Sec. II.D.] Simi-larly, a phase shift between the two logical states can beachieved by changing the gate voltage ng for some timeby a small amount, which modifies the energy differencebetween the ground and excited states.

Several remarks are in order:

(1) Unitary rotations by Bx and Bz are sufficient for allmanipulations of a single qubit. By using a sequenceof no more than three such elementary rotations wecan achieve any unitary transformation of a qubit’sstate.

(2) The example presented above, with control of Bzonly, provides an approximate spin flip for the situ-ation in which the idle point is far from degeneracyand EC@EJ . But a spin flip in the logical basis canalso be performed exactly. We must switch from theidle point h idle to the point where the effective mag-netic field is orthogonal to the idle one, h5h idle1p/2. This changes the Hamiltonian from H52 12 DE(h idle)rz to H52

12 DE(h idle1p/2)rx . To

achieve this, the dimensionless gate charge ngshould be increased by EJ /(4EC sin 2hidle). For thelimit discussed above, h idle!1, this operating point isclose to the degeneracy point, h5p/2.

(3) An alternative way to manipulate the qubit is to useresonant pulses, i.e., ac pulses with frequency closeto the qubit’s level spacing. We do not describe thistechnique here as it is well known from NMR meth-ods.

(4) So far we have been concerned with the time depen-dence during elementary rotations. However, fre-quently the quantum state should be kept un-changed for some time, for instance, while otherqubits are manipulated. Even in the idle state, h5h idle , because the energies of the two eigenstatesdiffer, their phases evolve relative to each other.This leads to coherent oscillations, typical for aquantum system in a superposition of eigenstates.We have to keep track of this time dependence withhigh precision and, hence, of the time t0 from thevery beginning of the manipulations. The time-dependent phase factors can be removed from theeigenstates if all the calculations are performed inthe interaction representation, with the zero-orderHamiltonian being the one at the idle point. How-ever, the price for this simplification is an additional


time dependence in the Hamiltonian during opera-tions, introduced by the transformation to the inter-action representation. This point has been discussedin more detail by Makhlin et al. (2000b).

(5) The choice of the qubit’s logical basis is by no meansunique. As follows from the preceding discussion,we can perform x and z rotations in the charge ba-sis, u↑& , u↓&, which provides sufficient tools for anyunitary operation. On the other hand, since we canperform any unitary transformation, we can chooseany other basis as a logical basis as well. The Hamil-tonian at the idle point is diagonal in the eigenbasis(2.7), while the controllable part of the Hamiltonian,the charging energy, favors the charge basis. Thepreparation procedure (thermal relaxation at theidle point) is more easily described in the eigenbasis,while coupling to the meter (see Sec. V) is diagonalin the charge basis. So the choice of the logical statesremains a matter of convention.

(6) A final comment concerns normal-metal single-electron systems. While they may serve as classicalbits and logic devices, they are ruled out as potentialquantum logic devices. The reason is that, due to thelarge number of electron states involved, their phasecoherence is destroyed in the typical sequential tun-neling processes.

B. Charge qubit with tunable coupling

A further step towards the ideal model (A1), in whichthe tunneling amplitude (x component of the field) iscontrolled as well, is the ability to tune the Josephsoncoupling. This is achieved by the design shown in Fig. 3,where the single Josephson junction is replaced by twojunctions in a loop configuration (Makhlin et al., 1999).This dc SQUID is biased by an external flux Fx , whichis coupled into the system through an inductor loop. Ifthe self-inductance of the SQUID loop is low (Tinkham,

FIG. 3. A charge qubit with tunable effective Josephson cou-pling. The single Josephson junction is replaced by a flux-threaded SQUID. The flux in turn can be controlled by acurrent-carrying loop placed on top of the structure.


1996), the SQUID-controlled qubit is described by aHamiltonian of the form (2.1) with modified potentialenergy:

2EJ0 cosS Q1p FxF0D2EJ0 cosS Q2p FxF0D522EJ

0 cosS p FxF0D cos Q . (2.10)Here F05hc/2e denotes the flux quantum. We assumethat the two junctions are identical4 with the same EJ

0 .The effective junction capacitance is the sum of indi-vidual capacitances of two junctions, in symmetric casesCJ52CJ

0 .When parameters are chosen such that only two

charge states play a role, we arrive again at the Hamil-tonian (2.3), but the effective Josephson coupling,

Bx5EJ~Fx!52EJ0 cosS p FxF0D , (2.11)

is tunable. Varying the external flux Fx by amounts oforder F0 changes the coupling between 2EJ

0 and zero.5

The SQUID-controlled qubit is thus described by theideal single-bit Hamiltonian (A1), with field componentsBz(t)5dEch@Vg(t)# and Bx(t)5EJ@Fx(t)# controlledindependently by the gate voltage and the flux. If we fixin the idle state Vg5Vdeg and Fx5F0/2, the Hamil-tonian is zero, Hqb0 50, and the state does not evolve intime. Hence there is no need to control the total timefrom the beginning of the manipulations, t0 . If wechange the voltage or the current, the modified Hamil-tonian generates rotations around the z or x axis, whichare elementary one-bit operations. Typical time spans ofsingle-qubit logic gates are determined by the corre-sponding energy scales and are of order \/EJ , \/dEchfor x and z rotations, respectively. If at all times at mostone of the fields, Bz(t) or Bx(t), is turned on, only thetime integrals of their profiles determine the results ofthe individual operations. Hence these profiles can bechosen freely to optimize the speed and simplicity of themanipulations.

The introduction of the SQUID not only permits sim-pler and more accurate single-bit manipulations, butalso allows us to control the two-bit couplings, as weshall discuss next. Furthermore, it simplifies the mea-surement procedure, which is more accurate at EJ50(see Sec. V).

C. Controlled interqubit coupling

In order to perform two-qubit logic gates we need tocouple pairs of qubits together and to control the inter-

4While this cannot be guaranteed with high precision in anexperiment, we note that the effective Josephson coupling canbe tuned to zero exactly by a design with three junctions.

5If the SQUID inductance is not small, the fluctuations of theflux within the SQUID renormalize the energy (2.10). But still,by symmetry arguments, at Fx5F0/2 the effective Josephsoncoupling vanishes.


actions between them. One possibility is to connect thesuperconducting boxes (i and j) directly, e.g., via a ca-pacitor. The resulting charge-charge interaction is de-scribed by a Hamiltonian of the form (A2) with an Ising-type coupling term }sz

i szj . Such a coupling allows easy

realization of a controlled-NOT operation. On the otherhand, it has severe drawbacks. In order to control thetwo-bit interaction, while preserving the single-bit prop-erties discussed above, one needs a switch to turn thetwo-bit interaction on and off. Any externally operatedswitch, however, connects the system to the dissipativeexternal circuit, thus introducing dephasing effects (seeSec. IV). They are particularly strong if the switch isattached directly to the qubit and unscreened, whichwould be required in order to control the direct capaci-tive interaction. Therefore alternatives were explored inwhich the control fields were coupled only weakly to thequbits. A solution (Makhlin et al., 1999) is shown in Fig.4. All N qubits are connected in parallel to a commonLC-oscillator mode that provides the necessary two-bitinteractions. It turns out that the ability to control theJosephson couplings by an applied flux simultaneouslyallows us to switch the two-bit interaction for each pairof qubits. This brings us close to the ideal model (A2)with a coupling term }sy

i syj .

In order to demonstrate the mentioned properties ofthe coupling we consider the Hamiltonian of the chain(register of qubits) shown in Fig. 4:

H5(i51

N H ~2eni2CgVgi!22~CJ1Cg! 2EJ~Fxi!cos Q iJ1

12NCqb

S q2 CqbCJ (i 2eniD2

1F2

2L. (2.12)

Here q denotes the total charge accumulated on thegate capacitors of the array of qubits. Its conjugate vari-able is the phase drop f across the inductor, related tothe flux by f/2p5F/F0 . Furthermore,

Cqb5CJCg

CJ1Cg(2.13)

FIG. 4. A register of many charge qubits coupled by oscillatormodes in the LC circuit formed by the inductor and the qubitcapacitors.


is the capacitance of the qubit in the external circuit.Depending on the relations among the parameters,

the Hamiltonian (2.12) can be reduced. We first considerthe situation in which the frequency of the (q ,F) oscil-lator, vLC

(N)51/ANCqbL , is higher than typical frequen-cies of the qubit’s dynamics:

\vLC(N)@EJ ,dEch . (2.14)

In this case the oscillator modes are not excited, but stilltheir virtual excitation produces an effective couplingbetween the qubits. To demonstrate this we eliminatethe variables q and F and derive an effective descriptionin terms of the qubits’ variables only. As a first step weperform a canonical transformation, q̃5q2(Cqb /

CJ) (2eni and Q̃ i5Q i12p(Cqb /CJ) (F/F0), while Fand ni are unchanged. This step leads to the new Hamil-tonian (we omit the tildes)

H5 q2

2NCqb1

F2

2L1(

iF ~2eni2CgVgi!22~CJ1Cg!

2EJ~Fxi!cosS Q i2 2pF0 CqbCJ F D G . (2.15)We assume that the fluctuations of F are weak,

CqbCJ

A^F2& ! F0 , (2.16)

since otherwise the Josephson tunneling terms in theHamiltonian (2.15) are washed out (Shnirman et al.,1997). Assuming Eq. (2.16) to be satisfied, we expandthe Josephson terms in Eq. (2.15) up to linear terms inF. Then we can trace over the variables q and F. As aresult we obtain an effective Hamiltonian, consisting ofa sum of N one-bit Hamiltonians (2.1) and the couplingterms

Hcoup522p2L

F02 S CqbCJ D

2F(i

EJ~Fxi!sin Q iG 2. (2.17)In spin-12 notation this becomes

6

Hcoup52(i,j

EJ~Fxi!EJ~Fxj!

ELŝy

i ŝyj1const, (2.18)

where we introduced the scale

EL5S CJCqbD2 F0

2

p2L. (2.19)

The coupling Hamiltonian (2.18) can be understoodas the magnetic free energy of the current-biased induc-tor 2LI2/2. This current is the sum of the contributionsfrom the qubits with nonzero Josephson coupling, I}( iEJ

i (Fxi)sin Qi}(iEJi (Fxi)ŝy

i .Note that the strength of the interaction does not de-

pend directly on the number of qubits N in the system.

6While expression (2.18) is valid only in leading order in anexpansion in EJ

i /\vLCN , higher terms also vanish when the Jo-

sephson couplings are put to zero. Hence the decoupling in theidle periods persists.


However, the frequency of the (q ,F) oscillator vLC(N)

scales as 1/AN . The requirement that this frequency notdrop below typical eigenenergies of the qubit ultimatelylimits the number of qubits that can be coupled by asingle inductor.

A system with flux-controlled Josephson couplingsEJ(Fxi) and an interaction of the form (2.18) allows usto perform all necessary gate operations in a straightfor-ward way. In the idle state all Josephson couplings areturned off and the interaction (2.18) is zero. Dependingon the choice of idle state we may also tune the qubitsby their gate voltages to the degeneracy points, whichmakes the Hamiltonian vanish, H50. The interactionHamiltonian remains zero during one-bit operations, aslong as we perform only one such operation at a time,i.e., for one qubit we have EJ

i 5EJ(Fxi)Þ0. To performa two-bit operation for any pair of qubits, say, i and j , EJ

i

and EJj are switched on simultaneously, yielding the

Hamiltonian

H52EJ

i

2ŝx

i 2EJ

j

2ŝx

j 2EJ

i EJj

ELŝy

i ŝyj . (2.20)

While Eq. (2.20) is not identical to Eq. (A2) it equallywell allows the relevant nontrivial two-bit operations,which, combined with the one-bit operations discussedabove, provide a universal set of gates.

A few comments should be added:

(1) We note that typical time spans of two-bit opera-tions are of the order \EL /EJ

2 . It follows from con-ditions (2.14) and (2.16) that the interaction energyis never much larger than EJ . Hence at best thetwo-bit gate can be as fast as a single-bit operation.

(2) It may be difficult to fabricate a nanometer-scaleinductor with the required inductance L , in particu-lar, since it is not supposed to introduce stray ca-pacitances. However, it is possible to realize such anelement by a Josephson junction in the classical re-gime (with negligible charging energy) or an arrayof junctions.

(3) The design presented above does not permit per-forming single- or two-bit operations simultaneouslyon different qubits. However, this becomes possiblein more complicated designs in which parts of themany-qubit register are separated, for example, byswitchable SQUID’s.

(4) In the derivation of the qubit interaction presentedhere we have assumed a dissipation-less high-frequency oscillator mode. To minimize dissipationeffects, the circuit, including the inductor, should bemade of superconducting material. Even so, at finitefrequencies some dissipation will arise. To estimateits influence, the effect of Ohmic resistance R in thecircuit has been analyzed by Shnirman et al. (1997),with the result that the interqubit coupling persists ifthe oscillator is underdamped, R!AL/NCqb. In ad-dition the dissipation causes dephasing. An estimateof the resulting dephasing time can be obtainedalong the lines of the discussion in Sec. IV. For a


reasonably low-loss circuit the dephasing due to thecoupling circuit is weaker than the influence of theexternal control circuit.

(5) The interaction energy (2.18) involves via EL theratio of CJ and Cqb . The latter effectively screensthe qubit from electromagnetic fluctuations in thevoltage source’s circuit, and hence should be takenas low as possible (see Sec. IV). Consequently, toachieve a reasonably high interaction strength andhence speed for two-bit operations, a large induc-tance is needed. For typical values of EJ;100 mKand Cg /CJ;0.1 one needs an inductance of L>1 mH in order not to have the two-bit operationmore than ten times slower than the single-bit op-eration. However, large values of the inductance aredifficult to reach without introducing large stray ca-pacitances. To overcome this problem Makhlin et al.(2000a) suggested using separate gate capacitors tocouple the qubits to the inductor, as shown in Fig. 5.As long as the superconducting circuit of the induc-tor is at most weakly dissipative, there is no need toscreen the qubit from the electromagnetic fluctua-tions in this circuit, and one can choose CL as largeas CJ (still larger CL would decrease the chargingenergy EC of the superconducting box), whichmakes the relevant capacitance ratio in Eq. (2.17) oforder one. Hence a fairly low inductance induces aninteraction of sufficient strength. For instance, forthe circuit parameters mentioned above, L;10 nHwould suffice. At the same time, potentially dephas-ing voltage fluctuations are screened by Cg!CJ .

(6) So far we have discussed manipulations on timescales much slower than the eigenfrequency of theLC circuit, which leave the LC oscillator perma-nently in the ground state. Another possibility is touse the oscillator as a bus mode, in analogy to thetechniques used for ion traps. In this case an ac volt-age with properly chosen frequency is applied to aqubit to entangle its quantum state with that of theLC circuit (for instance, by exciting the oscillatorconditionally on the qubit’s state). Then by address-ing another qubit one can absorb the oscillatorquantum, simultaneously exciting the second qubit.As a result, a two-qubit unitary operation is per-

FIG. 5. A register of charge qubits coupled to an inductor viaseparate capacitors CL;CJ , independent from the gate ca-pacitors Cg .


formed. This coupling via real excitations is a first-order process, as opposed to the second-order inter-action (2.18). Hence this method allows for fastertwo-qubit operations. Apart from this technical ad-vantage, the creation of entanglement between a qu-bit and an oscillator would by itself be a very inter-esting experimental achievement (Buisson andHekking, 2000).

D. Experiments with Josephson charge qubits

Several of the concepts and properties describedabove have been verified in experiments. This includesthe demonstration of superpositions of charge states, thespectroscopic verification of the spectrum, and even thedemonstration of coherent oscillations.

In a superconducting charge box the coherent tunnel-ing of Cooper pairs produces eigenstates that are gate-voltage-dependent superpositions of charge states. Thisproperty was first observed, in a somewhat indirect way,in the dissipative current through superconductingsingle-electron transistors. In this system single-electrontunneling processes (typically treated in perturbationtheory) lead to transitions between the eigenstates.Since the eigenstates are not pure charge states, theCooper-pair charge may also change in a transition. Inthe resulting combination of coherent Cooper-pair tun-neling and stochastic single-electron tunneling thecharge transferred is not simply e and the work done bythe voltage source not simply eV . [In an expansion inthe Josephson coupling to nth order the charge (2n11)e is transferred.] As a result a dissipative currentcan be transferred at subgap voltages. The theoreticalanalysis predicted a richly structured I-V characteristicat subgap voltages (Averin and Aleshkin, 1989; Maassenvan den Brink et al., 1991; Siewert and Schön, 1996),which has been qualitatively confirmed by experiments(Maassen van den Brink et al., 1991; Tuominen et al.,1992; Hadley et al., 1998).

A more direct demonstration of eigenstates that ariseas superpositions of charge states was found in theSaclay experiments (Bouchiat, 1997; Bouchiat et al.,1998). In their setup (see Fig. 6) a single-electron tran-sistor was coupled to a superconducting charge box (asin the measurement setup to be discussed in Sec. V) andthe expectation value of the charge of the box was mea-sured. When the gate voltage was increased adiabati-cally this expectation value increased in a series ofrounded steps near half-integer values of ng . At lowtemperatures the width of this transition agreed quanti-tatively with the predicted ground-state properties ofEqs. (2.3) and (2.7). At higher temperatures, the excitedstate contributed, again as expected from theory.

Next we mention the experiments of Nakamura et al.(1997), who studied the superconducting charge box byspectroscopic means. When exposing the system to ra-diation they found resonances (in the tunneling currentin a suitable setup) at frequencies corresponding to the


difference in the energy between excited and groundstates, again in quantitative agreement with the predic-tions of Eq. (2.3).

The most spectacular demonstration so far of the con-cepts of Josephson qubits has been provided by Naka-mura et al. (1999). Their setup is shown in Fig. 7. Inthese experiments the Josephson charge qubit was pre-pared far from the degeneracy point for a sufficientlylong time to relax to the ground state. In this regime theground state was close to a charge state, say, u↑& . Thenthe gate voltage was suddenly switched to a differentvalue. Let us first discuss the case in which it wasswitched precisely to the degeneracy point. Then the ini-tial state, a pure charge state, was an equal-amplitudesuperposition of the ground state u0& and the excitedstate u1&. These two eigenstates have different energies,hence in time they acquire different phase factors:

uc~ t !&5e2iE0 t/\u0&1e2iE1 t/\u1&. (2.21)

After a delay time Dt the gate voltage was switchedback to the original gate voltage. Depending on the de-lay, the system then ended up either in the ground state

FIG. 6. Scanning electron micrograph of a Cooper-pair boxcoupled to a single-electron transistor used in the experimentsof the Saclay group (Bouchiat, 1997; Bouchiat et al., 1998).

FIG. 7. Micrograph of a Cooper-pair box with a flux-controlled Josephson junction and a probe junction (Naka-mura et al., 1999).


u↑& [for (E12E0)Dt/h52np with n integer], in the ex-cited state u↓& [for (E12E0)Dt/h5(2n11)p], or ingeneral in a Dt-dependent superposition. The probabil-ity that, as a result of this manipulation, the qubit is inthe excited state is measured by monitoring the currentthrough a probe junction. In the experiments this cur-rent was averaged over many repeated cycles, involvingrelaxation and switching processes, and the oscillatorydependence on Dt described above was observed.

In fact even more details of the theory have beenquantitatively confirmed. For instance, one also expectsand finds an oscillatory behavior when the gate voltageis switched to a point different from the degeneracypoint, with the frequency of oscillations being a functionof this gate voltage. Second, the frequency of the coher-ent oscillations depends on the Josephson coupling en-ergy. The latter can be varied, since the Josephson cou-pling is controlled by a flux-threaded SQUID (see Fig.3). This aspect has also been verified quantitatively.

Coherent oscillations with a period of roughly 100 pscould be observed in the experiments of Nakamura et al.(1999) for at least 2 ns.7 This puts a lower limit on thephase coherence time tf and, in fact, represents its firstdirect measurement in the time domain. Estimates showthat a major contribution to the dephasing is the mea-surement process by the probe junction itself. In the ex-periments so far the detector was permanently coupledto the qubit and observed it continuously. Still, informa-tion about the quantum dynamics could be obtainedsince the coupling strength was optimized: it was weakenough not to destroy the quantum time evolution tooquickly and strong enough to produce a sufficient signal.A detector that does not induce dephasing during ma-nipulations should significantly improve the operation ofthe device. In Sec. V we suggest using a single-electrontransistor, which performs a quantum measurement onlywhen switched to a dissipative state.

So far only experiments with single qubits have beensucessfully carried out. Obviously the next step is tocouple two qubits and to create and detect entangledstates. Experiments in this direction have not yet beensuccessful, partially because of difficulties such as, forinstance, dephasing due to fluctuating backgroundcharges. However, the experiments using single qubitsimply that extensions to coupled qubits should be pos-sible as well.

E. Adiabatic charge manipulations

Another qubit design, based on charge degrees offreedom in Josephson-junction systems, was proposedby Averin (1998). It also allows control of two-bit cou-pling at the price of representing each qubit by a chainof Josephson-coupled islands. The basic setup is shownin Fig. 8. Each superconducting island (with index i) is

7In later experiments the same group reported phase coher-ence times as long as 5 ns (Nakamura et al., 2000).


biased via its own gate capacitor by a gate voltage Vi .The control of these voltages allows one to move thecharges along the chain analogously to the adiabaticpumping of charges in junction arrays (see, for example,Pekola et al., 1999). The capacitances of the Josephsonjunctions as well as the gate capacitances are smallenough so that the typical charging energy prevails overthe Josephson coupling. In this regime the appropriatebasis is that of charge states un1 ,n2 , . . . &, where ni is thenumber of extra Cooper pairs on island i . There existgate voltage configurations such that the two chargestates with the lowest energy are almost degenerate,while all other charge states have much higher energy.For instance, if all voltages are equal except for the volt-ages Vm and Vl at two sites, m and l , one can achievea situation in which the states u0,0,0, . . . & andu0, . . . ,21m,0, . . . ,1l , . . . & are degenerate. The subspaceof these two charge states is used as the logical Hilbertspace of the qubit. They are coupled via Josephson tun-neling across the um2lu21 intermediate junctions.

The parameters of the qubits’ Hamiltonian can betuned via the bias voltages. Obviously the bias energyBz(V1 ,V2 , . . . ) between these two states can bechanged via the local voltages Vl and Vm . Furthermore,the effective tunneling amplitude Bx(V1 ,V2 , . . . ) canbe tuned by adiabatic pumping of charges along thechain, changing their distance um2lu and hence the ef-fective Josephson coupling, which depends exponen-tially on this distance. (The Cooper pair must tunnel viaum2lu21 virtual charge states with much higher en-ergy.)

An interqubit interaction can be produced by placinga capacitor between the edges (outer islands) of two qu-bits. If at least one of the charges in each qubit is shiftedcloser to this capacitor, the Coulomb interaction leads toan interaction of the type Jzzsz

1sz2 . The resulting two-bit

Hamiltonian is of the form

H52 12 (j51,2 @Bz

j ~ t !szj 1Bx

j ~ t !sxj #1Jzz~ t !sz

1sz2 .

(2.22)

For controlled manipulations of the qubit the coeffi-cients of the Hamiltonian are modified by adiabatic mo-tion of the charges along the junction array. The adiaba-ticity is required to suppress transitions betweendifferent eigenstates of the qubit system.

While conceptually satisfying, this proposal appearsdifficult to implement: It requires many gate voltages foreach qubit. Due to the complexity a high degree of ac-curacy is required for the operation. Its larger size ascompared to simpler designs makes the system morevulnerable to dephasing effects, for example, due to

FIG. 8. Two coupled qubits as proposed by Averin (1998).


fluctuations of the offset charges.Adiabatic manipulations of the Josephson charge qu-

bit can lead to Berry phases. Falci et al. (2000) suggestedthat a Berry phase could accumulate during suitable ma-nipulations of a flux-controlled charge qubit with anasymmetric dc SQUID, and that it could be detected inan experiment similar to that of Nakamura et al. (1999).If the bare Josephson couplings of the SQUID loop areEJ

1 and EJ2 the effective Josephson energy is given by [cf.

Eq. (2.10)]

2EJ1 cosS Q1p FxF0D2EJ2 cosS Q2p FxF0D . (2.23)

Hence the corresponding Hamiltonian of the qubit hasall three components of the effective magnetic field: Bx5(EJ

11EJ2)cos(pFx /F0) and By5(EJ

22EJ1)sin(pFx /

F0), while Bz is given by Eq. (2.4). With three nonzerofield components, adiabatic changes of the control pa-rameters Vg and Fx may result in B’s enclosing a non-zero solid angle. This results in a Berry phase shift gBbetween the ground and excited states. In general, a dy-namic phase *DE(t)dt is also accumulated in the pro-cess. To single out the Berry phase, Falci et al. (2000)suggested encircling the loop in parameter space backand forth, with a NOT operation performed in between.The latter exchanges the ground and excited states, andas a result the dynamic phases accumulated during bothpaths cancel. At the same time the Berry phases add upto 2gB . This phase shift can be measured by a proce-dure similar to that used by Nakamura et al. (1999): thesystem is prepared in a charge state away from degen-eracy, abruptly switched to the degeneracy point whereadiabatic manipulations and the NOT gate are per-formed, and then switched back. Finally, the averagecharge is measured. The probability of finding the qubitin the excited charge state sin2 2gB reflects the Berryphase.

The experimental demonstration of topologicalphases in Josephson-junction devices would constitute anew class of macroscopic quantum effects in these sys-tems. They could be performed with a single Josephsonqubit in a design similar to that used by Nakamura et al.(1999) and thus appear feasible in the near future.

III. QUBITS BASED ON THE FLUX DEGREE OF FREEDOM

In the previous section we described the quantum dy-namics of low-capacitance Josephson devices where thecharging energy dominates over the Josephson energy,EC@EJ , and the relevant quantum degree of freedom isthe charge on superconducting islands. We shall now re-view the quantum properties of superconducting devicesin the opposite regime, EJ@EC , where the flux is theappropriate quantum degree of freedom. These systemswere proposed by Caldeira and Leggett (1983) in themid 1980s as test objects to study various quantum-mechanical effects, including macroscopic quantum tun-neling of the phase (or flux) as well as resonance tunnel-ing. Both had been observed in several experiments


(Voss and Webb, 1981; Martinis et al., 1987; Clarkeet al., 1988; Rouse et al., 1985). Another important quan-tum effect has been reported recently: The groups atStony Brook (Friedman et al., 2000) and Delft (van derWal et al., 2000) have observed in experiments theavoided level crossing due to coherent tunneling of theflux in a double-well potential. In principle, all othermanipulations discussed in the previous section shouldbe possible with Josephson flux devices as well. Theyhave the added advantage of not being sensitive to fluc-tuations in the background charges. However, attemptsto observe macroscopic quantum coherent oscillations inJosephson flux devices have not been yet successful(Leggett, 1987; Tesche, 1990).

A. Josephson flux (persistent current) qubits

We consider superconducting ring geometries inter-rupted by one or several Josephson junctions. In thesesystems persistent currents flow and magnetic fluxes areenclosed. The simplest design of these devices is an rfSQUID, which is formed by a loop with one junction, asshown in Fig. 9(a). The phase difference across the junc-tion is related to the flux F in the loop (in units of theflux quantum F05h/2e) by w/2p5F/F01integer. Anexternally applied flux Fx biases the system. Its Hamil-tonian, with Josephson coupling, charging energy, andmagnetic contributions taken into account, thus reads

H52EJ cosS 2p FF0D1 ~F2Fx!2

2L1

Q2

2CJ. (3.1)

Here L is the self-inductance of the loop and CJ thecapacitance of the junction. The charge Q52i\]/]Fon the leads is canonically conjugate to the flux F.

If the self-inductance is large, such that bL[EJ /(F0

2/4p2L) is larger than 1 and the externally ap-plied flux Fx is close to F0/2, the first two terms in theHamiltonian form a double-well potential near F5F0/2. At low temperatures only the lowest states inthe two wells contribute. Hence the reduced Hamil-tonian of this effective two-state system again has theform (2.3), Hctrl52 12 Bzŝz2 12 Bxŝx . The diagonal termBz is the bias, i.e., the asymmetry of the double-wellpotential, given for bL21!1 by

FIG. 9. The simplest flux qubits: (a) The rf SQUID, a simpleloop with a Josephson junction, forms the simplest Josephson

flux qubit; (b) improved design for a flux qubit. The flux F̃x inthe smaller loop controls the effective Josephson coupling ofthe rf SQUID.


Bz~Fx!54pA6~bL21 ! EJ ~Fx /F021/2!. (3.2)Bz can be tuned by the applied flux Fx . The off-diagonal term Bx describes the tunneling amplitude be-tween the wells, which depends on the height of the bar-rier and thus on EJ . This Josephson energy, in turn, canbe controlled if the junction is replaced by a dc SQUID,

as shown in Fig. 9(b), introducing the flux F̃x as anothercontrol variable.8 With these two external control pa-rameters the elementary single-bit operations, i.e., z andx rotations, can be performed, equivalent to the manipu-lations described for charge qubits in the previous sec-tion. In addition, for flux qubits we can either performthe operations by sudden switching of the external fluxes

Fx and F̃x for a finite time, or we can use ac fields andresonant pulses. To permit coherent manipulations theparameter bL should be chosen larger than unity (sothat two wells with well-defined levels appear) but notmuch larger, since the resulting large separation of thewells would suppress the tunneling.

The rf SQUID described above had been discussed inthe mid 1980s as a realization of a two-state quantumsystem. Some features of macroscopic quantum behav-ior were demonstrated, such as macroscopic quantumtunneling of the flux, resonant tunneling, and level quan-tization (Voss and Webb, 1981; Martinis et al., 1987;Clarke et al., 1988; Rouse et al., 1995; Silvestrini et al.,1997). However, only very recently has the level repul-sion near a degeneracy point been demonstrated (Fried-man et al., 2000; van der Wal et al., 2000).

The group at Stony Brook (Friedman et al., 2000)probed spectroscopically the superposition of excitedstates in different wells. The rf SQUID used had self-inductance L5240 pH and bL52.33. A substantialseparation of the minima of the double-well potential(of order F0) and a high interwell barrier made the tun-nel coupling between the lowest states in the wells neg-ligible. However, both wells contain a set of higher lo-calized levels—under suitable conditions one state ineach well—with relative energies also controlled by Fxand F̃x . Because they were closer to the top of the bar-rier, these states mixed more strongly and formed eigen-states, which were superpositions of localized flux statesfrom different wells. External microwave radiation wasused to pump the system from a well-localized loweststate in one well to one of these eigenstates. The energyspectrum of these levels was studied for different biases

Fx , F̃x , and the properties of the model (3.1) were con-firmed. In particular, the level splitting at the degen-eracy point indicated a superposition of distinct quan-tum states. They differed in a macroscopic way: theauthors estimated that the two superimposed flux statesdiffered in flux by F0/4, in current by 2–3 mA, and inmagnetic moment by 1010mB .

8See Mooij et al. (1999) for suggestions on how to control F̃xindependent of Fx .


The Delft group (van der Wal et al., 2000) performedmicrowave spectroscopy experiments on a similar butmuch smaller three-junction system, to be described inmore detail below. In their system the superpositions ofthe lowest states in two wells of the Josephson potentiallandscape were probed. In the spectrum they observed alevel repulsion at the degeneracy point, confirming thepredictions of the two-state model Hamiltonian (2.3)with the parameters Bx , Bz calculated from the poten-tial (3.3).

In spite of this progress, attempts to observe macro-scopic quantum coherence, i.e., the coherent oscillationsof a quantum system prepared in a superposition ofeigenstates, have not been successful so far (Leggett,1987; Tesche, 1990). A possible reason for this failurewas suggested recently by Mooij et al. (1999). They ar-gue that for the designs considered so far the existenceof the double-well potential requires that bL.1, whichtranslates into a rather high product of the critical cur-rent of the junction and its self-inductance. In practice,only a narrow range of circuit parameters is useful, sincehigh critical currents require a relatively large junctionarea resulting in a high capacitance, which suppressestunneling. A high self-inductance of the rf SQUID canbe achieved only in large loops. This makes the systemvery susceptible to external noise.

To overcome this difficulty Mooij et al. (1999) andFeigelman et al. (2000) proposed using a smaller super-conducting loop with three or four junctions, respec-tively. Here we discuss the three-junction circuit shownin Figs. 10(a) and (c). In this low-inductance circuit theflux through the loop remains close to the externallyapplied value, F5Fx . Hence the phase differencesacross the junctions are constrained by w11w21w352pFx /F0 , leaving w1 and w2 as independent dynami-cal variables. In the plane spanned by these two vari-ables the Josephson couplings produce a potential land-scape given by

U~w1 ,w2!52EJ cos w12EJ cos w2

2ẼJ cos~2pFx /F02w12w2!. (3.3)

If ẼJ /EJ.0.5, a double-well potential is formed withineach 2p32p cell in the phase plane. For an optimal

value of ẼJ /EJ'0.7–0.8 the cells are separated by highbarriers, while tunneling between two minima withinone cell is still possible. The lowest states in the wellsform a two-state quantum system, with two differentcurrent configurations. Mooij et al. (1999) and Orlandoet al. (1999) discuss junctions with EJ;2 K and EJ /EC;80 and loops of micrometer size with very small self-inductance L;5 pH (which can be neglected when cal-culating the energy levels). Typical qubit operation pa-rameters are the level splitting Bz;0.5 K and thetunneling amplitude Bx;50 mK. For the optimal choice

of ẼJ /EJ the two minima differ in phases by an amountof order p/2. Due to the very low inductance and therelatively low critical current Ic;200 nA, this translates


into a flux difference of dF;LIc;1023F0 . While this

corresponds to a still ‘‘macroscopic’’ magnetic momentof 104 –105mB , the two basis states are similar enough tomake the coupling to external fluctuating fields andhence the dephasing effects weak (for a further discus-sion, see Sec. IV). In this respect the new design is quali-tatively superior to the simple rf SQUID.

In order to obtain more direct evidence for superpo-sitions of localized flux states the Delft group (van derWal et al., 2000) measured the average flux of the qubitas a function of the external bias Bz . This experiment issimilar to that of Bouchiat et al. (1997, 1998) for thesingle-Cooper-pair box, which was discussed in Sec.II.D. As the bias is swept across the degeneracy point,one expects the average flux to change from the value inone well to that in the other well. At high temperaturesthe step is rounded, with width set by temperature. AskBT is lowered below the tunnel splitting Bx this widthshould saturate at the value of Bx . However, experi-mentally it saturated much earlier than expected fromthe spectroscopically measured tunnel splitting. This dis-crepancy indicates an enhanced population of the ex-cited state, which could be caused by noise, either fromexternal sources or due to the dc-SQUID detector.

FIG. 10. The Delft design of a flux qubit: (a) and (c) A three-junction loop as a flux qubit (Mooij et al., 1999). The reducedsize and lower inductance of this system as compared withearlier designs [e.g., Fig. 9(a)] reduce the coupling to the ex-ternal world and hence dephasing effects. (b) Multijunctionflux qubit with a controlled Josephson coupling (Mooij et al.,

1999). Control over two magnetic fluxes, F and F̃ , allows oneto perform all single-qubit logic operations.


B. Coupling of flux qubits

In order to couple different flux qubits one can use adirect inductive coupling (Mooij et al., 1999; Orlandoet al., 1999), as shown by the dashed line in Fig. 11. Amutual inductance between the qubits can be estab-lished in different ways. The dashed loop shown in thefigure couples the currents and fluxes in the lower partsof the qubits. Since fluxes through these loops controlthe barrier heights of the double-well potentials, thisgives rise to the interaction term }ŝx

1ŝx2 . Placing the

loop differently produces in addition contributions tothe interaction Hamiltonian of the form ŝz

1ŝz2 . The typi-

cal interaction energy is of order MIc2 where M is the

mutual inductance and Ic5(2p/F0)EJ is the critical cur-rent in the junctions. For their design, Mooij et al. (1999)estimate the typical interaction energy to be of order0.01EJ;50 mK in frequency units, i.e., of the order ofsingle-qubit energies. For a typical rf SQUID (Friedmanet al., 2000) this coupling can be even stronger than thetunneling rate between the flux states of the SQUID.

In the simplest form this interaction is always turnedon. To turn it off completely, one needs a switch con-trolled by high-frequency pulses. The related coupling tothe external circuit leads to decoherence (see the discus-sion at the end of this section). An alternative is to keepthe interaction turned on constantly and use ac drivingpulses to induce coherent transitions between the levelsof the two-qubit system (see Shnirman et al., 1997; Mooijet al., 1999). A disadvantage of this approach is that per-manent couplings result in an unwanted accumulation ofrelative phases between the two-qubit states even in theidle periods. Keeping track of these phases, or their sup-pression by repeated refocusing pulses (see Sec. IV), re-quires a high precision and complicates the operation.

A controllable interqubit coupling without additionalswitches is achieved in the design shown by the solid linein Fig. 11 (Makhlin et al., 2000c). The coupling is medi-ated by an LC circuit, with self-inductance Losc and ca-pacitance Cosc , which is coupled inductively to each qu-bit. Like the design of the charge qubit register in Sec.II.C, the coupling depends on parameters of individualqubits and can be controlled in this way. The effectivecoupling can be found again by integrating out the fastoscillations in the LC circuit. It can be understood in asimple way by noting that in the limit Cosc→0 the qubitsestablish a voltage drop across the inductor, V

FIG. 11. Flux qubits coupled in two ways. The dashed lineinduces a direct inductive coupling. Alternatively, an interqu-bit coupling is provided by the LC circuit indicated by a solidline.


5(iMḞi /L, and the Hamiltonian for the oscillator modeis Hosc5F2/2Losc1Q2/2Cosc2VQ , with the charge Qbeing conjugate to the flux F through the LC circuit.Here F i is the flux in the loop of qubit i , L is the self-inductance of the loop, and M is its mutual inductancewith the LC circuit. Continuing as described in Sec.II.C, we obtain the interqubit interaction term2CoscV

2/2. In the limit of weak coupling to the LC

circuit, we have Ḟ i5(i/\)@Hi ,F i#5dF iBxi ŝyi /\ , wheredF i is the separation between two minima of the poten-tial and Bx

i is the tunneling amplitude. Hence the inter-action is given by

Hint52p2S ML D2

(i,j

dF idF jF0

2

Bxi Bx

j

e2/Coscŝy

i ŝyj . (3.4)

To turn off the interaction one should suppress thetunneling amplitudes Bx

i . This can be done with expo-nential precision by increasing the height of the poten-

tial barrier via F̃x . Note that in this case unwanted fluc-tuations of Bx

i and resulting dephasing effects are alsoexponentially suppressed. All needed single and two-qubit manipulations can be performed by turning on thefields Bx

i and Bzi , in complete analogy to what we dis-

cussed in Sec. II.C. We also encounter the equivalentdrawbacks: the design shown in Fig. 11 does not allowsimultaneous manipulations on different qubit pairs, andthe conditions of high oscillator frequencies and weakrenormalization of qubit parameters by the coupling,similar to Eqs. (2.14) and (2.16), limit the two-qubit cou-pling energy. The optimization of this coupling requiresALosc /Cosc'RK(dF/F0)2(M/L)2 and vLC not farabove the qubit frequencies. For rf SQUID’s (Friedmanet al., 2000) the resulting coupling can reach the sameorder as the single-bit terms. On the other hand, for thedesign of Mooij et al. (1999), in which two basis phasestates differ only slightly in their magnetic properties,the coupling term is much weaker than the single-bitenergies.

C. ‘‘Quiet’’ superconducting phase qubits

The circuits considered so far in this section are vul-nerable to external noise. First, they need for their op-eration an external bias in the vicinity of F0/2, whichshould be kept stable for the time of manipulations. Inaddition, the two basis flux states of the qubit have dif-ferent current configurations, which may lead to mag-netic interactions with the environment and possiblecross talk between qubits. To a large extent the lattereffect is suppressed already in the design of Mooij et al.(1999). To further reduce these problems several designsof so-called ‘‘quiet’’ qubits have been suggested(Ioffe et al., 1999; Zagoskin, 1999; Blais and Zagoskin,2000; Blatter, 2001) They are based on intrinsically dou-bly degenerate systems, e.g., Josephson junctions withd-wave leads and energy-phase relations (e.g., cos 2f)with two minima, or the use of p junctions, which re-moves the need for a constant magnetic bias near F0/2.


The relevant two states differ only in their distributionof internal currents in the Josephson junctions while ex-ternal loops carry no current. As a result the coupling ofthe qubit to the electromagnetic environment is substan-tially reduced and coherence is preserved longer.

Designs have been proposed using tunnel junctionsbetween s- and d-wave superconductors (SD), betweentwo d-wave superconductors (DD), and between twod-wave superconductors with a normal-metal bridge(DND). The mentioned designs are similar and we dis-cuss them in parallel. Ioffe et al. (1999) suggested usingan SD tunnel junction with the s-wave lead matched tothe (110) boundary of the d-wave superconductor. Inthis geometry the first harmonic, }cos w, in the Joseph-son coupling vanishes due to symmetry reasons, and oneobtains a bistable system with the potential energyEJ cos 2w and minima at 6w0 with w05p/2. [A similarcurrent-phase relation was observed recently by II’ichevet al. (1998) in a DD junction with a mismatch angle of45°.] The DND design, with different orientations forthe two d-wave superconductors, was proposed byZagoskin (1999). The energy-phase relation for suchjunctions also has two degenerate minima, at the phasedifferences 6w0 . The separation 2w0 of these minima,and hence the tunneling amplitude, are controlled by themismatch angle of the d-wave leads.

In a later development a ‘‘macroscopic analog’’ ofd-wave qubits was discussed (Blatter et al., 2001). In-stead of an SD junction, it is based on a five-junctionloop, shown in Fig. 12, which contains one strong p junc-tion and four ordinary junctions. The presence of the pjunction is equivalent to magnetically biasing the loopwith a half superconducting flux quantum. Four otherjunctions, frustrated by the p phase shift, have twolowest-energy states with the phase difference of 6p/2between the external legs in the figure. In this respectthe five-junction loop is similar to the SD junction dis-cussed above and can be called a p/2 junction.

In all these designs the bistability is a consequence oftime-reversal symmetry (which changes the signs of allthe phases) of the Hamiltonian. Thus the degeneracyalso persists in systems containing different Josephsonjunctions, although the phase differences in the twolowest-energy states and their separation can change. Ifcharging effects with EC!EJ are included, one arrives ata double-well system with tunneling between the wells.Such a qubit can be operated by connecting or discon-

FIG. 12. A five-junction loop, a basic bistable element of‘‘quiet’’ superconducting qubits (Blatter et al., 2001), is madeof four ordinary junctions and one stronger p junction. In twostable configurations the phase difference across this elementis 6p/2.


necting it from external elements, as described below.The first issue to be addressed is how to store the

qubit’s state, i.e., how to freeze the evolution. This canbe achieved by connecting the qubit in parallel to a largecapacitor (Ioffe et al., 1999). This makes the phase de-gree of freedom very massive, thus suppressing the tun-neling and restoring the needed degeneracy. In order toperform a ŝx rotation, one turns on interwell tunnelingby disconnecting the capacitor. This means a switch isneeded in the circuit.

The ŝz rotation or phase shift can be accomplished bylifting the degeneracy between the wells. This can bedone by connecting another, much stronger p/2 junction(an SD-junction or five-junction loop) and a weak ordi-nary s-wave junction (with Josephson energy }cos w) inseries to the qubit, to form a closed loop. This againrequires a switch. The auxiliary p/2 junction shifts thephase differences of the potential minima of the qubit to0 and p. Hence the s junction is in the ground state orfrustrated depending on the qubit’s phase drop. The cor-responding energy difference produces the neededphase shift between two qubit’s states.

To perform two-qubit manipulations and control theentanglement, Ioffe et al. (1999) proposed forming aloop, connecting in series two qubits and one s junctionwith weak Josephson coupling EJ

s!EJ . The phase stateof each qubit is characterized by the phase difference of6w0 , i.e., the total phase drop on the qubits is equal to62w0 or 0 depending on whether the qubits are in thesame state or in different ones. When the connectionbetween the qubits and the s junction is turned on, thisphase drops across the s junction, and its energy differsby EJ

s(12cos 2w0) for the states u00& , u11& as comparedto the states u01&, u10&. The net effect is an Ising-typeinteraction between the pair of qubits, which allows uni-tary two-qubit transformations.

Another mode of operation was discussed by Blaisand Zagoskin (2000). They suggested using a magneticforce microscope tip for single-bit manipulations (localmagnetic field lifts the degeneracy of two phase states)and for the readout of the phase state. The tip should bemoved towards or away from the qubit during manipu-lations. The short time scales of qubit operation makethis proposal difficult to realize.

Even in ‘‘quiet’’ designs, in both SD and DD systems,there are microscopic persistent currents flowing insidethe junctions which differ for the two logical states (Za-goskin, 1999; Blatter et al., 2001). These weak currentsstill couple to the outside world and to other qubits, thusspoiling the ideal behavior. Furthermore, all the designsmentioned require externally operated switches to con-nect and disconnect qubits. We discuss the associatedproblems in the following subsection.

To summarize, the quiet designs require rather com-plicated manipulations as well as circuits with manyjunctions, including p junctions or d-wave junctions,which are difficult to fabricate in a controlled and reli-able way. In addition, many constraints imposed on thecircuit parameters (in particular, on the hierarchy of Jo-sephson couplings) appear difficult to satisfy. In our


opinion the quiet phase qubit designs belong to a higher-complexity class than the previously discussed chargeand flux qubits, and their experimental realization mayremain a challenge for some time.

D. Switches

Switches may be used in a variety of contexts in quan-tum nanocircuits. They are needed, for example, for adirect capacitive coupling between charge qubits ormagnetic coupling of flux qubits. They are also a majortool for controlling the dynamics of quiet qubits. Idealswitches should decouple qubits from the environmentand at the same time let through control signals. Theyshould operate on the very fast time scale of the qubitdynamics and have a high switching ratio, that is, theratio of the interaction with the switch in the on or offstate. Such switches are hard to realize. In this sectionwe compare the characteristics of several Josephson-junction-based switches and their associated problems.

Possible switches are dc SQUID’s as well as SSET’s(superconducting single-electron transistors, or single-Cooper-pair transistors) in a mode in which they act asJosephson junctions with an externally controlled cou-pling. Then the switching ratio is the ratio of the mini-mal and maximal values of the coupling. In a dc SQUIDwith Josephson energies of its junctions equal to EJ

1 andEJ

2 , this ratio is (EJ12EJ

2)/(EJ11EJ

2). It reached a valuebelow 1% in the experiment of Rouse et al. (1995).However, fast switching of the bias flux may be difficultto perform. In a SSET the effective coupling is con-trolled by a gate voltage, which can be switched quickly.However, the switching ratio of order EJ /EC (EJ andEC are characteristics of the SSET) is hardly below sev-eral percent. These limitations lead to unwanted inter-actions when the switch is supposed to be disconnected.

Since a dc SQUID requires an external bias to beoperated as a switch, Blatter et al. (2001) suggested asimilar construction with the bias provided by p/2 junc-tions instead of an external magnetic field. That is, onecould insert two p/2 junctions into one arm of theSQUID loop. Depending on whether the phase dropsacross these junctions were equal or opposite, theywould simulate an external bias of a half flux quantumor no bias. Accordingly, the Josephson couplings of twos junctions in the SQUID would add up or cancel eachother. The switching could be realized via a voltagepulse that drives one p/2 junction between 1p/2 and2p/2 states. Blatter et al. (2001) also suggested using anarray of n such switches, reducing the overall Josephsoncoupling in the off state by a factor @(EJ

12EJ2)/EC#

n.Unfortunately, in the on state the overall couplingthrough the array would also be reduced with growingn , although this reduction might be weaker than in theoff state, i.e., the switching ratio increases with n . Nev-ertheless the quality of the switch in the on state wouldbe reduced. Moreover, to operate the switch one wouldneed to send voltage pulses simultaneously to n interme-diate elements, which complicates the operation. Notethat this design is reminiscent of the qubit design pro-


posed by Averin (1998), which is presented in Sec. II.E.However, while Blatter et al. (2001) suggest controlingthe coupling, }(EJ /EC)

n by controlling EJ , Averin pro-poses to changing the distance n of the tunneling pro-cess.

While switches of the type described above may beuseful in first experiments with simple quantum nanocir-cuits, further work is needed before they can be used inmore advanced designs that require high precision ofmanipulations and phase coherence over a long periodof time.

IV. ENVIRONMENT AND DISSIPATION

A. Identifying the problem

For an ideal quantum system the time evolution isdescribed by deterministic, reversible unitary opera-tions. The concepts of quantum-state engineering andcomputation heavily rely on this quantum coherence,with many potential applications requiring a large num-ber of coherent manipulations of a large number of qu-bits. On the other hand, for any real physical quantumsystem the time evolution may be disturbed in variousways, and the number of coherent manipulations is lim-ited. Possible sources of error are inaccuracies in thepreparation of the initial state, inaccuracies in the ma-nipulations (logic gates), uncontrolled couplings be-tween qubits, undesired excitations out of the two-stateHilbert space (Fazio et al., 1999), and—unavoidable indevices that are to be controlled externally—interactions with the environment. Due to the couplingto the environment, the quantum state of the qubits getsentangled with the environmental degrees of freedom.As a consequence the phase coherence is destroyed af-ter a time scale called the dephasing time. In this sectionwe shall describe the influence of the environment onthe qubit. We determine how the dephasing time de-pends on system parameters and how it can be opti-mized.

Some of the errors can be corrected by software tools.One known from NMR and, in particular, NMR-basedquantum logic operations (see, for example, Chang,1998) is refocusing. Refocusing techniques serve to sup-press the effects of undesired terms in the Hamiltonian,e.g., deviations of the single-bit field terms from theirnominal values or uncontrolled interactions like straydirect capacitive couplings of charge qubits or inductivecouplings of flux qubits. As an example we consider theerror due to a single-bit term dBxsx , which after sometime has produced an unwanted rotation by a. Refocus-ing is based on the fact that a p pulse about the z axisreverses the influence of this term, i.e.,Uz(p)Ux(a)Uz(p)5Ux(2a). Hence fast repeated in-versions of the bias Bz(t) (with uBzu@dBx) eliminatethe effects associated with dBx . The technique can alsobe applied to enhance the precision of nonideal controlswitches: one first turns off the coupling term to a lowvalue and then further suppresses it by refocusing. Theexamples demonstrate that refocusing requires very fast


repeated switchings with a period much shorter than theelementary operation time. This can make it hard toimplement.

It was therefore a major breakthrough when the con-cepts of quantum error correction were discovered (see,for example, Preskill, 1998; Steane, 1998). When appliedthey should make it possible, even in the presence ofdephasing processes—provided that the dephasing timeis not too short—to perform coherent sequences ofquantum manipulations of arbitrary length. The price tobe paid is an increase in system size (by roughly an orderof magnitude), and a large number of steps are neededfor error correction before another computational stepcan be performed (increasing the number of steps byroughly three orders of magnitude). This imposes con-straints on the dephasing time. Detailed analysis showsthat error correction can be successful if the dephasingtime is of the order of 104 times longer than the timeneeded for an elementary logic gate.

In the Josephson-junction systems discussed here, theenvironment is usually composed of resistive elements inthe circuits needed for the manipulations and the mea-surements. They produce voltage and current noise. Inmany cases such fluctuations are Gaussian distributedwith a Johnson-Nyquist power spectrum, coupling lin-early to the quantum system of interest. They can thusbe described by a harmonic oscillator bath with suitablefrequency spectrum and coupling strength (Leggettet al., 1987; Weiss, 1999). For charge qubits, for instance,fluctuations in the gate voltage circuit, coupling to sz ,and fluctuations in the current, which control the Jo-sephson energy and couple to sx , can be described inthis way (Shnirman et al., 1997). In this section we shallfirst describe these noise sources and the dephasing thenintroduce. We later comment on other noise sourcessuch as telegraph noise, typically with a 1/f power spec-trum due to switching two-level systems (e.g., back-ground charge fluctuations), or the shot noise resultingfrom tunneling in a single-electron transistor coupled toa qubit for the purpose of a measurement.

Depending on the relation between typical frequen-cies of the coherent (Hamiltonian) dynamics and thedephasing rates, we distinguish two regimes. In the first,the Hamiltonian-dominated regime, where the con-trolled part of the qubit Hamiltonian Hctrl52(1/2)Bs,governing the deterministic time evolution and logicgates, is large, it is convenient to describe the dynamicsin the eigenbasis of Hctrl . The coupling to the environ-ment is weak, hence the environment-induced transi-tions are slow. One can then distinguish two stages: (a)dephasing processes, in which the relative phase betweenthe eigenstates becomes random; and (b) energy relax-ation processes, in which the occupation probabilities ofthe eigenstates change.

In the second, environment-dominated, regime Hctrl istoo weak to support its eigenstates as the preferred ba-sis. The qubit’s dynamics in this situation is governed bydissipative terms and depends on details of the structureof the coupling to the environment. In general the evo-


lution is complicated, and the distinction between relax-ation and dephasing may be impossible.

Both regimes may be encountered during manipula-tions. Obviously the Hamiltonian should dominate whena coherent manipulation is performed. On the otherhand, if in the idle state the Hamiltonian vanishes (avery useful property as outlined in Secs. II.A and II.B),the environment-dominated regime is realized. One hasto ensure that the phase coherence rate in this regime isstill low enough.

B. Spin-boson model

Before we proceed to discuss specific physical sys-tems, let us review what is known about the spin-bosonmodel, which has been studied extensively (see reviewsby Leggett et al., 1987 and Weiss, 1999). It models theenvironment as an oscillator bath coupled to one com-ponent of the spin. The Hamiltonian reads

H5Hctrl1sz(a

laxa1HB , (4.1)

where

Hctrl5212

Bz sz212

Bx sx (4.2)

52DE

2~cos h sz1sin h sx! (4.3)

is the controlled part of the Hamiltonian [cf. Eqs. (2.3)and (2.6)], while

HB5(a

S pa22ma 1 mava2xa

2

2 D (4.4)is the Hamiltonian of the bath. The bath operator X5(alaxa couples to sz . In thermal equilibrium onefinds for the Fourier transform of the symmetrized cor-relation function of this operator

^Xv2 &[

12 ^

$X~ t !,X~ t8!%&v5\J~v!cothv

2kBT, (4.5)

where the bath spectral density is defined by

J~v![p

2 (ala

2

mavad~v2va!. (4.6)

This spectral density typically has a power-law behaviorat low frequencies (Leggett et al., 1987). Of particularinterest is Ohmic dissipation, corresponding to a spec-trum

J~v!5p

2a\v , (4.7)

which is linear at low frequencies up to some high-frequency cutoff vc . The dimensionless parameter a re-flects the strength of dissipation. Here we concentrateon weak damping, a!1, since only this regime is rel-evant for quantum-state engineering. But still theHamiltonian-dominated and the environment-


dominated regimes are both possible depending on theratio between the energy scale DE5ABz21Bx2, charac-terizing the coherent evolution, and the dephasing rate(to be determined below).

The Hamiltonian-dominated regime is realized whenDE@akBT . In this regime it is natural to describe theevolution of the system in the eigenbasis (2.7) which di-agonalizes Hctrl :

H52 12

DErz1~sin h rx1cos h rz! X1HB . (4.8)

Two different time scales characterize the evolution(Görlich et al., 1989; Weiss and Wollensak, 1989; Weiss,1999). On a first, dephasing time scale tw the off-diagonal (in the preferred eigenbasis) elements of thequbit’s reduced density matrix decay to zero. They arerepresented by the expectation values of the operatorsr6[(1/2)(rx6iry). Dephasing leads to the followingtime dependence (at long times):

^r6~ t !&5^r6~0 !&e7iDEte2t/tw. (4.9)

On the second, relaxation time scale trelax the diagonalentries tend to their thermal equilibrium values:

^rz~ t !&5rz~`!1@rz~0 !2rz~`!#e2t/trelax, (4.10)

where rz(`)5tanh(DE/2kBT).The dephasing and relaxation times were originally

evaluated for the spin-boson model in a path-integraltechnique (Leggett et al., 1987; Weiss, 1999). The ratesare9

trelax21 5pa sin2 h

DE

\coth

DE

2kBT, (4.11)

tw215

12

trelax21 1pa cos2 h

2kBT\

. (4.12)

In some cases these results can be derived in a simpleway, which we present here to illustrate the origin ofdifferent terms. As is apparent from the Hamiltonian(4.8) the problem can be mapped on the dynamics of aspin-1/2 particle in the external magnetic field DE point-ing in the z direction and a fluctuating field in the x-zplane. The x component of this fluctuating field, withmagnitude proportional to sin h, induces transitions be-tween the eigenstates (2.7) of the unperturbed system.Applying the golden rule for this term, one obtainsreadily the relaxation rate (4.11).

The longitudinal component of the fluctuating field,proportional to cos h, does not induce relaxation pro-cesses. It does, however, contribute to dephasing since itleads to random fluctuations of the eigenenergies andthus to a random relative phase between the two eigen-

9Note that in the literature usually the evolution of ^sz(t)&has been studied. To establish the connection to the results(4.11) and (4.12) one has to substitute Eqs. (4.9) and (4.10)into the identity sz5cos h rz1sin h rx . Furthermore, we ne-glect renormalization effects, since they are weak for a!1.


states. As an example we analyze its effect on thedephasing rate in an exactly solvable limit.

The unitary operator

U[expS 2isz F2 D with F[(a 2lapa\mava2 (4.13)transforms the Hamiltonian (4.1)–(4.4) to a rotating spinframe (Leggett et al., 1987):

H̃5UHU2152~1/2!DE cos h sz2~1/2!DE sin h~s1e

2iF1H.c.!1HB . (4.14)Here we recognize that in the limit h50 the spin andthe bath are decoupled, which allows an exact treat-ment. The trivial time evolution in this frame, s6(t)5exp(7iDEt)s6(0), translates in the laboratory frameto

s6~ t !5e7iF(t)e6iF(0)e7iDEts6~0 !. (4.15)

To average over the bath we need the correlator

P~ t ![^eiF(t)e2iF(0)&5^e2iF(t)eiF(0)&, (4.16)

which has been studied extensively by many authors(Leggett et al., 1987; Odintsov, 1988; Panyukov andZaikin, 1988; Nazarov, 1989; Devoret et al., 1990). It canbe expressed as P(t)5exp@K(t)#, where

K~ t !54

p\ E0`

dvJ~v!

v2

3FcothS \v2kBT D ~cos vt21 !2i sin vtG . (4.17)For the Ohmic bath (4.7) for t.\/2kBT one hasRe K(t)'2(2kBT/\)p a t . Thus we reproduce Eq. (4.9)with tw given by Eq. (4.12) in the limit h50. While it isnot so simple to derive the general result for arbitrary h,it is clear from Eqs. (4.11) and (4.12) that the effects ofthe perpendicular (}sin h) and longitudinal (}cos

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Quantum-state engineering with Josephson-junction devicesfy.chalmers.se/~delsing/QI/Makhlin-RMP-01.pdf · 2003. 1. 15. · Josephson junction or the ﬂux in a ring geometry near